I'm totally confused by recurrence relations. We just learned about relations, I don't even see the relation part... so solving them is not coming easily to me at all. In fact everything I see or read in my textbook or around the Internet is just confusing me more because they all use different terms and notation and random letters for variables that I don't even know what they're supposed to represent. Very frustrated.
Here's an extremely simple one, I would so appreciate if someone could attempt to explain to me how to obtain the solution, and how the same principle is supposed to apply to other linear homogeneous recurrences with constant coefficients, as I know there's supposedly some trick to those particular types. Something like $$ t_{n} = S^n $$ but what is the $t$ and what is the $S$?? How do I get those?
Here's the relation to solve: $$ a_{n} = -3a_{n-1}, a_{0} = 2$$
I've seen it done like super simply in that this would be $$ t^n = -3^n$$ but I know that's not correct, because the answer given is $$ a_{n} = 2(-3)^n$$
So where does the 2 come from? Unless it comes from the initial condition, in which case sometimes there are multiple (e.g., a0 and a1), so then what?
Thank you.
$ a_1=−3a_{0} $
$ a_2=−3a_{1} $
$ \cdots $
$ a_n=−3a_{n−1} $
Multiply all these equations and simplify to get $a_n=(-3)^n a_0$.